Scrub and debeard mussels; discard any that are not closed.
In a large stockpot over medium heat add oil, then green onions, garlic, shallots, and pepper. Cook until tender, about 3 minutes, then add wine and mussels. Bring to a boil, reduce heat to medium, and steam, covered, for 3 to 5 minutes (until mussels open).
With a slotted spoon, transfer mussels to a large dish. Discard any unopened mussels. Cover with plastic wrap to keep warm.
Strain liquid through a fine sieve, gently pressing to extract as much juice as possible. Return liquid to pan. Stir in mustard and cream and place pan over medium heat. Bring to a boil, then reduce heat and stir until slightly thickened. Add parsley; season to taste with sea salt and pepper.
To serve, place mussels back in stockpot or divide mussels among four heated bowls and top with sauce. Garnish with parsley. Serve immediately with baguettes.
Tip- The mussels’ beard, known as the byssus thread to marine biologists, doesn’t appear on every mussel. If you see some frizzy threads where the shells meet, simply pinch them and tug steadily, with a side- to-side motion, to remove them. And don’t worry if you leave a few behind—they aren’t harmful.